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2006 The algebraic crossing number and the braid index of knots and links
Keiko Kawamuro
Algebr. Geom. Topol. 6(5): 2313-2350 (2006). DOI: 10.2140/agt.2006.6.2313

Abstract

It has been conjectured that the algebraic crossing number of a link is uniquely determined in minimal braid representation. This conjecture is true for many classes of knots and links.

The Morton–Franks–Williams inequality gives a lower bound for braid index. And sharpness of the inequality on a knot type implies the truth of the conjecture for the knot type.

We prove that there are infinitely many examples of knots and links for which the inequality is not sharp but the conjecture is still true. We also show that if the conjecture is true for K and , then it is also true for the (p,q)–cable of K and for the connect sum of K and .

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Keiko Kawamuro. "The algebraic crossing number and the braid index of knots and links." Algebr. Geom. Topol. 6 (5) 2313 - 2350, 2006. https://doi.org/10.2140/agt.2006.6.2313

Information

Received: 28 April 2006; Accepted: 21 July 2006; Published: 2006
First available in Project Euclid: 20 December 2017

zbMATH: 1128.57007
MathSciNet: MR2286028
Digital Object Identifier: 10.2140/agt.2006.6.2313

Subjects:
Primary: 57M25
Secondary: 57M27

Rights: Copyright © 2006 Mathematical Sciences Publishers

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